Shrink-wrap not conforming to X-Axis
Shrinkwrap is doing its best, with the provided vertices.. it can't bend an edge defined by only two points.


Subdivide the edges you want to conform to the curvature
Adjust the offset


And.. if you want a less destructive way (fussy clients, etc.), assign a lattice deform modifier to some or all of the vertices of the square, text, etc. and Shrinkwrap the lattice. This allows an extrusion without applying the Shrinkwrap.